STRUCTURES: PURPOSE AND FUNCTION
1.8 STRESS AND STRAIN
produce maximum stress conditions and maximum deformations. In ad- dition, the external loads often occur in different combinations, with each combination producing different internal force effects. This frequently makes the analysis of structural behaviors for design a quite laborious process, making us now very grateful for the ability to utilize computer- aided procedures in design work.
strain for a number of different materials. The form of such a graph illus- trates various aspects of the nature of structural behavior of the materials.
Curves 1 and 2 represent materials with a constant proportionality of the stress and strain magnitudes. For these materials, a quantified rela- tionship between stress and strain can be described simply in terms of the slope or angle of the straight line graph. This relationship is commonly expressed as the tangent of the angle of the graph and is called the modulus of elasticityof the material. The higher the value of this modu- lus—that is, the steeper the slope of the graph—the stiffer the material.
Thus, the material represented by curve 1 in the illustration is stiffer than the material represented by curve 2.
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Figure 1.35 Direct stress and strain.
Figure 1.36 Shear stress and strain.
For direct stress of tension or compression, the strain is measured as a linear change, and the modulus is called the direct stress modulus of elas- ticity. For shear stress, the strain is measured as an angular change, and the resulting modulus is called the shear modulus of elasticity.
Some materials, such as glass and very high-strength steel, have a constant modulus of elasticity for just about the full range of stress up to failure of the material. Other materials, such as wood, concrete, and plas- tic, have a curved form for the stress-strain graph (curve 3 in Figure 1.37). The curved graph indicates that the value for the modulus of elas- ticity varies continuously for the full range of stress.
The complex shape of curve 4 in Figure 1.37 is the characteristic form for a so-called ductilematerial, such as low-grade steel of the type ordi- narily used for beams and columns in buildings. This material responds elastically at a low level of stress, but suddenly deforms excessively at a level of stress described as its yield point. However, fracture does not usually occur at this level of stress, but rather at a higher level after the material reaches a certain limiting magnitude of yielding strain. This pre- dictable yield phenomenon and the secondary reserve strength are used to predict ultimate load capacities for steel frames, as well as for concrete structures that are reinforced with ductile steel rods.
Figure 1.37 Stress and strain relationships.
Stress Combinations
Stress and strain are three-dimensional phenomena, but for simplicity, they are often visualized in linear or planar form. As shown in Figure 1.35, direct stress of compression in a single direction results in strain of shortening of the material in that direction. However, if the volume of the material remains essentially unchanged—which it usually does—there will be a resulting effect of lengthening (or pushing out) at right angles to the compression stress. This implies the development of a tension ef- fect at right angles to the compression, which in some materials may be the real source of failure, as is the case for tension-weak concrete and plaster. Thus, a common form of failure for concrete in compression is by lateral bursting at right angles to the compression load.
If direct stress is developed in a linear member, as shown in Figure 1.38, the pure direct stress occurs only on sections at right angles to the direct force loading, called cross sections. If stress is considered on a sec- tion at some other angle (called an oblique section), there will be a com- ponent of shear on the section. If the material is weak in shear (such as wood parallel to its grain), this angular shear stress effect may be more critical than the direct stress effect.
Although simple linear tension and compression forces produce di- rect, linear stresses, shear stress is essentially two-dimensional, as shown in Figure 1.39. The direct effect of a shear force is to produce shear stresses that are parallel to the force (on faces aandbin Figure 1.39a).
These opposed stresses in the material produce a rotational effect, which must be balanced by other opposed stresses (at faces canddin Figure 1.39b). Thus, whenever shear stress exists within a structure, there is al- ways an equal magnitude of shear stress at right angles to it. An example
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Figure 1.38 Stress on a cross section not at right angles to the active force.
of this is the stack of loose boards used as a beam, as shown in Figure 1.27. The shear failure in this case is a horizontal slipping between the boards, even though the shear force is induced by vertical loading.
As shown in Figures 1.39candd, the combination of the mutually per- pendicular shear stresses produces a lengthening of the material on one diagonal and a shortening on the other diagonal. This implies the devel- opment of tension on one diagonal and compression on the other diago- nal, at right angles to the tension. In some cases, these diagonal stresses may be more critical than the shear stresses that produce them. In con- crete, for example, failure due to shear stress is usually actually a diago- nal tension stress failure, as this is the weakest property of the material.
Figure 1.39 Effects of shear.
On the other hand, high shear in the web of a steel beam may result in di- agonal compression buckling of the thin web.
Separately produced direct stresses in a single direction may be summed algebraically at a given point in a structure. In the case of the column shown in Figure 1.40, the compression load produces a direct compression stress on a cross section, as shown at Figure 1.40a, if the load is placed so as not to produce bending. If the load is off-center on the column, the stress conditions will be modified by the addition of bending stresses on the cross section, as shown in Figure 1.40b. The true net stress condition at any point on the cross section will thus be the simple addition of the two stress effects, with a combined stress distribution possible as shown in Figure 1.40c.
A more complex situation is the combination of direct stresses and shear stresses. Figure 1.41ashows the general condition at a point in the cross section of a beam where the net stress consists of a combination of the di- rect stress due to bending (tension or compression) and shear stress. These stresses cannot simply be added as they were for the column. What can be combined are the direct stress due to bending and the direct diagonal stress due to shear, as shown in Figure 1.41b. Actually, because there are two di- agonal stress conditions, there will be two combinations—one producing a maximum effect and the other a minimum effect, as shown in Figure 1.41c.
These two stress limits will occur in mutually perpendicular directions.
There is also a net combined shear stress, as shown in Figure 1.41d.
This is the combination of the direct shear stress and the diagonal shear stress due to the direct stress. Since the direct shear stress is at right an- gles (vertically and horizontally) and the shear stress due to direct stress is on a 45° plane, the net maximum shear will be at some angle between these two. This angle will be closer to a right angle when the direct shear is larger and closer to a 45° position when the direct stress is larger.
Another stress combination is that produced by triaxial stress condi- tions. An example of this is a confined material subjected to compression, such as air or liquid in a piston chamber, as shown in Figure 1.42. In addi- tion to being compressed by the active compressing force (the piston), the material is squeezed laterally by the other material around it. The net effect on the confined material is a three-way push, or triaxial compression. For materials with little or no tension resistance, such as air, water, or dry sand, this is the only situation in which they can resist compression. Thus, a sandy soil beneath a footing can develop resistance in the form of vertical soil pressure because of the confinement of the soil around it and above it.
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For visualization purposes, it is common to reduce complex structural actions to their component effects. These simpler individual effects can thus be analyzed more clearly and simply, and the results combined with the effects of the other components. In the end, however, care must be taken to include all the components for a given situation.
Figure 1.40 Combined direct stresses.
Thermal Stress
The volumes of materials change with temperature variation, increasing as temperatures rise and decreasing when they fall. This phenomenon creates a number of problems that must be dealt with in building design.
The form of objects determines the basic nature of significant di- mensional changes. As shown in Figure 1.43, the critical directions of
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Figure 1.41 Combined shear stress and direct stress.
Figure 1.42 Development of stress in a confined material.
Figure 1.43 Effects of thermal change on solid objects.
movement depend on whether the object is essentially linear, planar (two-dimensional), or three-dimensional. For a linear object (beam, col- umn, etc.), the significant change is in its length; significant concerns are those for very long objects, especially in climates with a considerable temperature range.
Planar objects, such as wall panels and large sheets of glass, expand in a two-dimensional manner. Attachments and constraints by other con- struction must allow for thermal movements. Three-dimensional move- ments are mostly dealt with by providing for component movements of a linear or two-dimensional nature.
If thermal expansion or contraction is resisted, stresses are produced.
Figure 1.44 shows a linear structural member in which length change is
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Figure 1.44 Effect of thermal change on a constrained element.
constrained. If the temperature is raised, the member will push outward against the restraints, developing internal compression as the constraints push back. This results in an external compression force on the member, in the same manner as a load applied to a column. With quantified val- ues known for the thermal expansion coefficient and the stress-strain re- lationship for the material, the compressive stress developed in the member can be determined.
Another type of thermal problem is that involving differential move- ment of attached parts of the construction. Figure 1.45 shows a common situation in which a cast concrete structure consists of elements of dif- ferent mass or thickness. If exposed to temperature change, the thinner parts will cool down or warm up more quickly than the thicker parts to which they are attached by the continuous casting process. The result is that the thinner parts are restrained in their movements by the thicker parts, which induces stresses in all the parts. These stresses are most crit- ical for the thinner parts and at the joints between the parts.
Another problem of differential thermal movements occurs between the exterior surface and the interior mass of a building. As shown in Figure 1.46, the exposed skin—as well as any exposed structural mem- bers—will tend to move in response to the changes in outdoor tempera- tures, while the interior elements of the construction tend to remain at a relatively constant, comfort-level temperature. For a multistory build- ing, this effect accumulates toward the top of the building and can result in considerable distortions in the upper levels of the structure.
A similar problem occurs with long buildings in which the part above ground is exposed to the weather, while that buried in the ground remains at a relatively constant temperature throughout the year (see Figure 1.47).
Figure 1.45 Critical stress effects resulting from differential thermal movements.
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Figure 1.46 Effect of exposure conditions of the structure on development of thermally induced stress and strain. (a) Conditions resulting in major exposure of the exterior wall structure, but enclosure of the interior structure. (b) In the winter (outside at 0°F, interior at 70°F, differential of 70°F), the exterior columns become shorter than the interior, resulting in the deformations shown. (c) In the summer (outside at 100°F, inside at 75°F, differential of 25°F), the exterior columns become longer than the interior, resulting in the deformations shown.
Figure 1.47 Thermal effects in partly underground buildings.
Composite Structures
When structural elements of different stiffness share a load, they develop resistance in proportion to their individual stiffnesses. As shown in Fig- ure 1.48a, if a group of springs share a load that shortens all of the springs the same amount, the portion of the load resisted by the stiffer springs will be greater, since it takes a greater effort to shorten them.
Another common type of composite structure occurs when concrete is reinforced with steel rods, as shown in Figure 1.48b. When a load is ap- plied to such an element (called a composite structure), the stiffer mate- rial (steel in this case) will carry a higher portion of the load. In this
Figure 1.48 Load sharing in composite structures. (a) A group of springs of varying stiffness. (b) Steel-reinforced concrete.
manner, a relatively small percentage of steel in a reinforced concrete member can be made to carry a major part of the load, since steel has on average around 10 times the stiffness of structural grade concrete.
A situation somewhat similar to this occurs when the building as a whole is distorted by loads, such as the horizontal effects of wind and earthquakes. Figure 1.49 shows two examples of this, the first being a building with solid walls of masonry and wood frame construction in the same exterior surface. As a bracing wall for horizontal loads, the much stiffer masonry will tend to take most of the load. In this case, the wood framed wall may be virtually ignored for its structural resistance, al- though any effects of the lateral distortion must be considered.
The second example in Figure 1.49 involves a steel frame in the same plane as relatively stiff walls. Even though the framed walls may be less strong than the steel frame, they will likely be much stiffer; thus, they will tend to absorb a major portion of the lateral load. The solution in this case is to either make the walls strong enough for the bracing work, or to make the steel frame stiff enough to protect the walls and actually do the bracing work.
Time-Related Stress and Strain
Some stress and strain phenomena are time related. Concrete is subject to an effect called creep (see Figure 1.50), in which the material sustains a progressive deformation when held at a constant stress over a long time.
These deformations are added to those produced normally by the initial
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Figure 1.49 Load sharing by elements of different construction.
loading. Additionally, unlike the initial deformations, they remain per- manent, similar to the long-term sag of wood beams.
Creep does not affect the stress resistance of concrete, but does result in some redistribution of stresses between the concrete and its steel rein- forcing. Since the steel does not creep, it effectively becomes increas- ingly stiffer in relation to the progressively softening concrete. This makes the steel even greater in its capability of carrying a major part of the load in the composite structure.
Soft, wet clay soils are subject to a time-related flow effect, similar to the slow oozing of toothpaste from a tube as it is squeezed. If the soil mass is well constrained (similar to putting the cap back on the toothpaste tube), this effect can be arrested. However, as long as there is some- where for the clay to ooze toward, and the pressure on it is maintained, the flow will continue. Instances of buildings that continue to settle over many years have occurred with this soil condition (see Figure 1.51).
Another time-related stress problem occurs when structures are re- peatedly loaded and unloaded. The effect of people walking, of wind and earthquakes, and of machinery rocking on its supports are cases of this loading condition in buildings. Some materials may fail from the fa- tigue effects of such loadings. However, a more common problem is that of loosening of connections or the progressive development of cracks that were initially created by other effects.
Figure 1.50 Effect of creep.